RL unknotter, hard unknots and unknotting number

TL;DR

This paper introduces a reinforcement learning approach to simplify knot diagrams, successfully tackling complex unknot diagrams and improving bounds on unknotting numbers.

Contribution

It presents a novel RL pipeline for knot simplification, capable of handling arbitrary knots and systematically improving unknotting number bounds.

Findings

Successfully applied to very hard unknot diagrams

Recovered the upper bound of three for the unknotting number of a complex knot

Proposed a self-improving extension to enhance bounds on prime knots

Abstract

We develop a reinforcement learning pipeline for simplifying knot diagrams. A trained agent learns move proposals and a value heuristic for navigating Reidemeister moves. The pipeline applies to arbitrary knots and links; we test it on ``very hard'' unknot diagrams and, using diagram inflation, on $4_1\#9_{10}$ where we recover the recently established and surprising upper bound of three for the unknotting number. In addition, we explain a self-improving workbook-driven extension of the pipeline that systematically improves unknotting number upper bounds on the list of prime knots.